A100069 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*4^(n-2*k).

1, 4, 18, 76, 326, 1384, 5892, 25036, 106438, 452344, 1922588, 8170936, 34726940, 147589264, 627256088, 2665837516, 11329815878, 48151714264, 204644809932, 869740430056, 3696396920116, 15709686864304, 66766169526008, 283756220309176, 1205963937666076, 5125346734404784

Offset: 0

Author

Paul Barry, Nov 02 2004

Keywords

• easy
• nonn

Comments

An inverse Chebyshev transform of x/(1-4*x), where the Chebyshev transform of g(x) is ((1-x^2)/(1+x^2))*g(x/(1+x^2)) and the inverse transform maps a g.f. A(x) to (1/sqrt(1-4*x^2))*A(x*c(x^2)) where c(x) is the g.f. of the Catalan numbers A000108. In general, Sum_{k=0..floor(n/2)} binomial(n,k) * r^(n-2*k) has g.f. 2*x/(sqrt(1-4*x^2)*(r*sqrt(1-4*x^2) + 2*x - r)). - corrected by Vaclav Kotesovec, Dec 06 2012

Generally (for r>1), a(n) ~ (r + 1/r)^n. - Vaclav Kotesovec, Dec 06 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A027306, A100067, A100068.

Programs

Magma

m:=4; [(&+[Binomial(n,k)*m^(n-2*k): k in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Jun 08 2022

Mathematica

CoefficientList[Series[x/(Sqrt[1-4*x^2]*(2*Sqrt[1-4*x^2]+x-2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 06 2012 *)

PARI

my(x='x+O('x^66)); Vec(x/(sqrt(1-4*x^2)*(2*sqrt(1-4*x^2)+x-2))) \\ Joerg Arndt, May 12 2013

SageMath

m=4; [sum(binomial(n,k)*m^(n-2*k) for k in (0..n//2)) for n in (0..40)] # G. C. Greubel, Jun 08 2022

Formula

G.f.: x/(sqrt(1-4*x^2)*(2*sqrt(1-4*x^2)+x-2)). - corrected by Vaclav Kotesovec, Dec 06 2012

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*4^(n-2*k).

a(n) = Sum_{k=0..n} binomial(n, (n-k)/2)*(1 + (-1)^(n-k))*4^k/2.

8*n*a(n) = 2*(19*n-4)*a(n-1) + (15*n+2)*a(n-2) - 8*(19*n-23)*a(n-3) + 68*(n-3)*a(n-4) = 0. - R. J. Mathar, Nov 22 2012

a(n) ~ 17^n/4^n. - Vaclav Kotesovec, Dec 06 2012